The Principles of Boolean Algebra

The fundamental properties of Boolean algebra simplify Boolean expressions and enhance the performance of systems that utilize Boolean logic.

Here are the main properties of Boolean operations:

1. Idempotence
   Repeating the same operation with the same operand and operator, whether AND or OR, leaves the result unchanged.
    \( A \land A = A \)
    \( A \lor A = A \)

   Example. For \( A = 1 \), we observe: $$ A \land A = 1 \land 1 = 1 $$ $$ A \lor A = 1 \lor 1 = 1 $$ This demonstrates that idempotence ensures the result remains the same when the same operation is applied to \( A \).

2. Associativity
   This property allows the re-grouping of operations without affecting the outcome, which facilitates the integration of multiple operations.
   \( (A \land B) \land C = A \land (B \land C) \)
   \( (A \lor B) \lor C = A \lor (B \lor C) \)

   Example. Setting \( A = 1 \), \( B = 0 \), and \( C = 1 \), we see associativity in action: $$ (A \land B) \land C = (1 \land 0) \land 1 = 0 \land 1 = 0 $$ $$ A \land (B \land C) = 1 \land (0 \land 1) = 1 \land 0 = 0 $$ Thus, the sequence of grouping \( A \) and \( B \) or \( B \) and \( C \) first does not impact the final result.

3. Commutativity
   The sequence of operands in an operation does not affect the outcome.
   \( A \land B = B \land A \)
   \( A \lor B = B \lor A \)

   Example. With \( A = 1 \) and \( B = 0 \), the outcomes show: $$ A \land B = 1 \land 0 = 0 $$ $$ B \land A = 0 \land 1 = 0 $$ This proves that commutativity ensures the result is independent of the order of the operands.

4. Identity Element
   The element "true" (1) acts as an identity for AND, while "false" (0) serves as the identity for OR.
   \( A \land 1 = A \)
   \( A \lor 0 = A \)

   Example. With $ A = 1 $: $$ A \land 1 = 1 \land 1 = 1 $$ $$ A \lor 0 = 1 \lor 0 = 1 $$ This illustrates how the identity element preserves the value of \( A \) in each operation.

5. Complement Law
   Combining an element $ A $ with its negation $ \neg A $ results in false (0) for AND and true (1) for OR.
   \( A \land \neg A = 0 \)
   \( A \lor \neg A = 1 \)

   Example. If \( A \) is set to 1: $$ A \land \neg A = 1 \land 0 = 0 $$ $$ A \lor \neg A = 1 \lor 0 = 1  $$ This shows that an element and its complement produce predictable results in Boolean operations.

6. De Morgan's Laws
   These fundamental rules help simplify complex Boolean expressions, especially when negating combined expressions.
   \( \neg (A \land B) = \neg A \lor \neg B \)
   \( \neg (A \lor B) = \neg A \land \neg B \)

   Example. For \( A = 1 \) and \( B = 0 \): $$ \neg (A \land B) = \neg (1 \land 0) = \neg 0 = 1 $$ $$ \neg (A \lor B) = \neg (1 \lor 0) = \neg 1 = 0 $$ This demonstrates how De Morgan's Laws convert negations into equivalent, more manageable expressions.

7. Distributivity
   This property allows for the expansion of expressions in ways analogous to traditional algebra.
   \( A \land (B \lor C) = (A \land B) \lor (A \land C) \)
   \( A \lor (B \land C) = (A \lor B) \land (A \lor C) \)

   Example. With \( A = 1 \), \( B = 1 \), and \( C = 0 \): $$ A \land (B \lor C) = 1 \land (1 \lor 0) = 1 \land 1 = 1 $$ $$ A \lor (B \land C) = 1 \lor (1 \land 0) = 1 \lor 0 = 1 $$ This highlights that whether expanding or condensing the operations, the result remains equivalent, mirroring distribution in classical mathematics.